PROCEEDINGS THE INSTITUTION OF ELECTRICAL ENGINEERS Volume 115 Power Unbalanced magnetic pull in a 6-pole induction motor M. Bradford, M.Sc. Synopsis A 6-pole JOkW induction motor has been specially constructed for a comprehensive series of steady-state and transient measurements of the magnitude of unbalanced magnetic pull resulting from an eccentric air gap. These measurements have shown that unbalanced magnetic pull is critically dependent on saturation of the magnetic circuit, and that the magnitude in the cage-rotor motor is much less than in the wound-rotor motor. Comparisons between measurements and calculations show that most existing theories inaccurately predict the magnitude of transient unbalanced magnetic pull during starting, the effect of slotting and saturation, the effect of a cage rotor, and the action of load currents. A simple theory is developed which gives reasonable agreement with measurements for the wound-rotor motor over a range of supply voltages about the rated value. List of symbols A = rotor surface area B, b = magnetic flux density D = rotor diameter F = unbalanced magnetic pull g = air gap length / = rotor length M = m.m.f. p = slot pitch R = reduction factor for slotting r = radius of rotor s = slot opening ⬠= eccentricity = 8fg a = magnetic force per unit area H0 = permeability of free space, 4TT X 10~ 7 /JL = permeability of magnetic material S = displacement between rotor and stator centres ijj = mechanical angle 1 Introduction In the design of an electrical machine, some dimen- sional tolerance is allowed on each separate part to simplify the manufacture of the machine. As a result, the centres of the rotor and stator will not be perfectly aligned. Hence, the machine will have a nonuniform air-gap length, and, if the windings are series-connected, a corresponding nonuni- form air-gap flux-density distribution. This will cause an unbalanced magnetic pull (u.m.p.), which is usually in the direction of the greatest air-gap flux density, i.e. the smallest air gap. The possible magnitude of this pull is required to. calculate the stiffness of the mechanical assembly, in par- ticular the shaft, necessary to limit the deflection caused by the pull to a safe value. If the shaft is too flexible, the deflection can cause rubbing between the rotor and stator. If the shaft is too stiff, the physical dimensions and weight of the machine will be excessive. The magnitude of the u.m.p. is also a decid- ing factor in the size of bearings required to transmit this radial force. If the u.m.p. is too great, the bearing size required Paper 5671 P, first received 23rd April and in revised form 16th July 1968 Mr. Bradford is with the Electrical Research Association, Leatherhead, Surrey, England PROC. 1EE, Vol. 115, No. 11, NOVEMBER 1968 75 P29 can make the design and construction of the machine unneces- sarily complicated. It may be desirable, for these reasons, to try to limit the magnitude of the u.m.p. by judicious design. The object of the present work was to determine the dependence of the u.m.p. on supply voltage, eccentricity and load, for motors with series-connected and parallel-connected rotors, i.e. wound-rotor and cage-rotor motors, respectively. The investigation was carried out at the Electrical Research Association, and is described in more detail elsewhere.1 This paper has been limited to the more important aspects. The possible benefits to be obtained from such an investiga- tion are considerable. An improvement in the design of 2-pole induction motors of about 1500kW rating, resulting from a better knowledge of u.m.p., has been previously described.2 It was possible to reduce the shaft diameter by 28%. Owing to the increased area of rotor core and lower flux density, and the addition of cooling ducts in the rotor, the output of the motor could then be increased by 50% for the same frame size. Conversely, to maintain the initial output, the stator diameter could be reduced by 15%, and the weight by 33%. The test motor is mounted vertically above a d.c. loading generator. The eccentricity is adjusted by positioning the stator horizontally with pairs of wedges. Radial forces between rotor and stator are recorded by force transducers arranged in the two bearing housings. The positions of the force trans- ducers ensure that any radial force between the rotor and stator is transmitted by the transducers. Flux densities are measured in the stator iron by search coils and, in the air- gap, by printed-circuit search conductors. These latter have a total thickness of only 0-06mm, and are spaced at 1 â¢27 mm intervals. The main dimensions etc. of the machine are shown in Table 1. Table 1 MACHINE DETAILS Stator outside diameter . Rotor outside diameter Core length Air-gap length . Stator slots Rotor slots. Rotor skew 400mm 230 mm 150 mm 0-75mm 54 36 0 1619 2 Steady-state measurements Measurements were made under steady-state conditions with a wound rotor, a cage rotor and an unslotted rotor. For each test condition, deflection of the rotor-stator assembly by u.m.p. was measured by change of capacitance across the air gap. Thus, any such deflection was allowed for in deter- mining the test eccentricity. Measurements were taken at the rated voltage of the machine, 415 V, and at 80% and 125% of rated voltage, 332V and 518 V, respectively. 2.1 Unbalanced magnetic pull The measured variations of u.m.p. with eccentricity, voltage and load, for both the wound-rotor and cage-rotor motors, are shown in Fig. 1. The steady-state values with the 3 1 10 5020 30 40 eccentricity, °/o Fig. 1 Variation of u.m.p. with eccentricity for 6-pole motors a No load b Full load ⢠518V x 415V O 332V (j) Wound rotor (li) Cage rotor unslotted rotor are not shown, but were approximately 10% greater than the no-load u.m.p. in the wound-rotor motor. The difference between the wound-rotor and cage-rotor motors is immediately noticeable, the u.m.p. in the latter being between 16% and 30% of the former. This reduction in u.m.p. results from equalising currents set up in the cage by the nonuniform distribution of air-gap flux density. The air-gap flux density is approximately inversely proportional to the air-gap length, and hence varies around the machine when the rotor and stator are eccentric. However, the rotor equalising currents are, by Lenz's law, in such a direction as to oppose the fields which establish them, and hence main- tain a more uniform distribution of air-gap, flux density. Consequently the u.m.p. is reduced. 2.1.1 Variation with eccentricity For all three rotors and all test conditions of load and voltage, the measured values of u.m.p. were directly propor- tional to eccentricity up to the experimental limit of about 45 % eccentricity. This is contrary to expectations of u.m.p. increasing more rapidly than the eccentricity above 20% eccentricity. For example, Freise and Jordan3 derive a 1620 multiplying factor for u.m.p. equal to 1 ⢠05 at 20 % eccentricity and 1-31 at 45% eccentricity. Such predictions ignore the effect of saturation prominent in the small air gap, which will limit the rise of u.m.p. One test was performed increasing the eccentricity to 65 %. For this, the wound-rotor motor was energised at 415 V, with the rotor open-circuited. The absence of rotation then allowed the insertion of a thin piece of paper in the small air gap to ensure that the rotor and stator surfaces did not touch. The test showed that the u.m.p. was still approximately directly proportional to eccentricity even at such high eccentricities, although a slight curvature of the graph indicated that the u.m.p. increased less rapidly than eccentricity. 2.1.2 Variation with voltage Assuming no saturation and an air-gap flux density proportional to line voltage, u.m.p. would then be directly proportional to the square of the line voltage. Hence, the 25% voltage increases from 332 to 415 V and from 415 to 518 V would correspond to 56% increases in u.m.p. The actual increases in u.m.p. were 28% and 7%, respectively, for the wound-rotor motor, and 55% and 62% for the cage-rotor motor, both motors being on no load. Thus, the u.m.p. in the cage-rotor motor increased approximately with the square of the line voltage, whereas, in the wound-rotor motor, the increase in u.m.p. was limited by saturation. The cage-rotor motor was run on no load over a range of voltages from 100 V up to 500 V, the results confirming the square-law agreement between u.m.p. and line voltage. This behaviour is surprising in that the magnetisation curves for the cage- rotor and wound-rotor motors show saturation to be greater in the former. However, owing to the equalising currents in the cage-rotor winding, the increase of flux density in the small air gap, for a given eccentricity, is much less than in the wound-rotor motor. Thus, the level of saturation is more uniform around the cage-rotor motor. This indicates the limiting factor, deciding the relationship between u.m.p. and voltage, to be local saturation in the small air gap, rather than the overall saturation level of the machine. With the unslotted rotor, u.m.p. increased by 34% and 24% for the two-25% increases in supply voltage. These increases are greater than those measured in the wound-rotor motor on no load. This is to be expected, as the absence of rotor teeth lowers the flux density in the rotor, and hence reduces the effect of saturation. Figs. 2 and 3 show the results of tests made on the wound-rotor 0-9 go-8 x 0) o O - 7 in 8,0 o â¢o 0 3 §,0 Gii) 6 8 line current,A 10 12 14 Fig. 2 Results of open-circuit-rotor test on 6-pole wound-rotor motor Nominal eccentricity = 1 2 % (i) Flux density in small air gap (ii) Flux density in large air gap (iii) Unbalanced magnetic pull (arbitrary scale) (iv) Voltage PROC. IEE, Vol. 115, No. 11, NOVEMBER 1968 motor, with the rotor winding open-circuited. The line-voltage/line-current curve is equivalent to the magnetisa- tion curve for the motor, and is linear to a line voltage of 2-4 2-2 §2-0 a1-8 I 1-4 o 10 |o-8 0-6 0-4 0-2 In(ii) / * 415V 518V x / 332V 100 200 WO 600300 400 line voltage ,V Fig. 3 Variation of u.m.p. with voltage for 6-pole wound-rotor motor Nominal eccentricity = 12%, rotor open-circuit (i) Measured u.m.p. (ii) U.M.P. proportional to voltage squared (theoretical) 350 V. The air-gap flux-density curves are, naturally, similar to the magnetisation curve, each flux density increasing linearly with line current up to O-55T. The u.m.p. is initially proportional to the square of the line current, while the magnetisation curve is linear. As the line current increases beyond the 'knee' of the magnetisation curve, the u.m.p. rapidly reaches a maximum value, and just begins to decrease at the maximum line current. Fig. 3 shows the variation of u.m.p. with line voltage, with the initial square-law relation- ship continued at higher voltages for comparison with the measured u.m.p. This illustrates well how rapidly the u.m.p. reaches a maximum above the 'knee' of the magnetisation curve at 350V. At voltages greater than the test values, the flux densities in the large and small air gaps will become more nearly equal, and hence u.m.p. will decrease still further. Tn the limit, the distribution of air-gap flux density will be almost independent of eccentricity, and the u.m.p. will approach zero. 2.1.3 Variation with load The u.m.p., measured when the wound-rotor motor was on load, was slightly greater than on no load. The ratios of u.m.p. on load to that on no load were 111, 105 and 102 for 332, 415 and 518V, respectively. Similarly, in the cage-rotor motor, the u.m.p. on load was greater than on no load, the ratios being 2 05, 1 -37 and 1 03 for 332, 415 and 518 V respectively. These results, from the wound-rotor and cage-rotor motors, were obtained with constant generator loading at each test voltage. Hence, the load current depended on the supply voltage. The cage-rotor motor was further tested at 45% eccentricity at each test voltage for a range of generator loadings from no load to overload. At each test condition, the u.m.p., line current and power input were measured. Each power factor was then calculated, and a circle-diagram technique was used to calculate the vector current due to the load. Fig. 4 shows the variation of u.m.p. with load current at each voltage. The increase of u.m.p. with load current is equivalent to a reduction in the effect of rotor-winding equalising currents, the increase being most pronounced at 332V and much less so at 518 V. At 332V, saturation is negligible, and the equalising currents are governed mostly by the load current. However, at 518 V, equalising currents are already limited by saturation, even at no load, and hence increase of load current only slightly increases u.m.p. PROC. 1EE, Vol. 115, No. 11, NOVEMBER 1968 2.2 Air-gap flux-density distribution Typical no-load and full-load air-gap flux-density distri- butions are shown in Fig. 5. On no load, the flux density is constant across a tooth, decreasing to about half this value over a slot. On full load, this distribution changes radically to an approximately linear decrease in flux density from the 1-4L 10 load current,A Fig. 4 Variation of u.m.p. with load current for 6-pole cage-rotor motor % ⢠S18V x 415V O 332V 1-1 r- 0-9 0-7 0-5 0-3 1 0 a 0-8 0-4 0-2 tooth [slot| I Ii li tooth slot | tooth 1 3 5 7 9 11 13 15 17 19 21 search-conductor number Fig. 5 Air-gap flux-density distribution for 6-pole wound-rotor motor e = 0 a No load b Full load ⢠518V x 415V O 332V 1621 leading edge of the tooth to the trailing edge. The percentage rise in flux density above the no-load value is approximately equal to the percentage decrease. This variation is due to load harmonics, and is consequently most pronounced at reduced voltage, when the full-load current is greatest. 2.3 Variation of air-gap flux density with eccentricity The percentage change of air-gap flux density with eccentricity is shown in Fig. 6, for both the wound-rotor and 3#/. -20 - * -20 -20 eccentricity,°/o Fig. 6 Percentage change of air-gap flux density with eccentricity on no load, for 6-pole motors a 332 V b 415V c 518V ⢠wound rotor, small air gap x wound rotor, large air gap A cage rotor, small air gap O cage rotor, large air gap Percentages denote change in air-gap flux density for 10% change in eccentricity cage-rotor motors. This shows the increase in flux density in the small air gap and the corresponding decrease in the large air gap. These percentage changes are all directly pro- portional to eccentricity, up to the experimental limit of about 45% eccentricity. The change of flux density with eccentricity in the cage-rotor motor was considerably less than in the wound-rotor motor, owing to the action of the parallel paths in the cage-rotor winding. This effect is most noticeable at 332 V, where saturation is negligible At 518 V, saturation reduces the difference between the wound-rotor and cage- rotor motors. 3 Transient measurements The transient condition studied was the period during and immediately after switching the motor on to the supply. The point-on-wave of switching was controlled in steps of 30°, from 60° to 240° after zero voltage on one phase. The wound-rotor motor was normally started with the rotor winding short-circuited. 3.1 General nature of transient u.m.p. A typical record of transient u.m.p. in the cage-rotor motor is shown in Fig. 7. This record was obtained with an ultraviolent recorder. The initial u.m.p. alternates for a few cycles at the mechanical resonant frequency, about a steady transient direct component. Then, the acceleration of the rotor sets up a component of u.m.p. at the rotor-slot fre- quency; i.e. the rotational speed multiplied by the number of rotor slots. The initial resonance rapidly decays, while the frequency of the rotor-slot component increases with speed. Around frequencies of 112 and 140Hz, 187 and 233rev/min, respectively, this component excites further resonant vibra- tion of the rotor-stator assembly. As the frequency increases, these resonances disappear, leaving only a component of u.m.p. alternating at a frequency corresponding to the speed of the motor, owing to a small rotating component of eccentricity. These alternating components are superimposed on direct components of u.m.p., a constant value during the transient starting period and a second constant value during the steady-state no-load running condition. In the cage-rotor motor, the transient u.m.p. was between four and five times greater than the no-load value. Records from the wound- rotor motor were very similar, differing mainly on two points. First, the initial oscillations occurred at line frequency, rather than at the mechanical resonant frequency, and, secondly, the transient u.m.p. was only about 40% greater than the no-load value. The measurements from the records showed no variation of transient u.m.p. with the point-on-wave of switching. Little significance can be attached to the magnitude of the initial peaks of u.m.p., as these are greatly dependent on the mechanical response of the motor and the electrical response of the measuring apparatus. 233rev/min 1 2 3 4 5 6 8 1012 141618 20 24 28 32 36 »_ 112HZâ 187rev/min (iii) ' ⢠' _i i i i i i_ 1 2 3 4 5 6 7 8 1012 14161820 24 28 32 I Fig. 7 Transient u.m.p. in 6-pole cage-rotor motor e = 45%, 332V (i) U.M.P. at the lower end of the motor (ii) U.M.P. at the upper end of the motor, showing observed rotor-slot positions (iii) Photoelectric-tachometer output showing calculated rotor-slot positions 1622 PROC. JEE, Vol. 115, No. II, NOVEMBER J968 3.2 Direct component of transient u.m.p. Fig. 8 shows the transient u.m.p., recorded with the cage-rotor motor, compared with the line current during the starting and no-load-running periods. The transient u.m.p. the end plates would be greater, and that the mass of the rotor would be less, resulting in a higher natural frequency. How- ever, it is still possible that noise and vibration could occur from this cause in the operating range of a commercial (iii) 100 200 300 400 500 600 700 800 900 Fig. 8 Relation between transient u.m.p. and starting current for 6-pole cage-rotor motor e = 45%, 332 V (i) U.M.P. at the lower end of the motor (ii) U.M.P. at the upper end of the motor (iii) Speed, rev/min (iv) Line current can be clearly seen to be virtually constant during the whole starting period, while the line current is equal to the high starting value. As the rotor speed approaches the no-load value, both the transient u.m.p. and the starting current rapidly reduce to their no-load values, much less than the transient values. The transient u.m.p. and starting current approximately maintain their standstill values up to about 800rev/min. Records from the wound-rotor motor show a very similar result, but, as the no-load value of u.m.p. is much larger than that of the cage-rotor motor, the reduction of u.m.p. from the transient starting condition to the no-load condition is far less dramatic. The variation of transient u.m.p. with eccentricity is shown in Fig. 9, in comparison with full-load values, for the wound- rotor and cage-rotor motors. As in the steady state, transient u.m.p. is directly proportional to eccentricity, except above 25% eccentricity in the cage-rotor motor, when the u.m.p. increases less rapidly. 3.3 Alternating component of transient u.m.p. Fig. 7 shows a record of transient u.m.p. in the cage- rotor motor, at 332V with 45% eccentricity, and the output of a photoelectric tachometer. The latter marks off successive revolutions of the rotor from the instant of switching. The rate of acceleration was found to be constant initially, and hence the first revolution has been subdivided into 36 divisions corresponding to each rotor slot. Counting back- wards from the end of the first revolution, the fundamental frequency of variation of u.m.p. has been marked out. Higher-frequency harmonics slightly obscure the pattern, but the correspondence between the calculated rotor-slot position and each cycle of variation of u.m.p. can easily be seen. This rotor-slot-frequency variation of u.m.p. excites a mechanical resonance in the bottom bearing housing at 112Hz at a rotational speed of 187rev/min and at 140Hz, 233rev/min in the top bearing housing. The difference between the mechanical resonant frequencies is due to the different and complex mechanical arrangement at each end of the rotor shaft. The variation of transient u.m.p. in the wound-rotor motor was very similar to that in the cage-rotor motor. The resonance in the test motor corresponded basically to the natural frequency of vibration of the force transducers when loaded by the mass of the rotor. In a comparable commercial machine, it is likely that the overall stiffness of PROC. IEE, Vol. 7/5, No. II, NOVEMBER 1968 machine. The degree of buildup of these resonant vibrations will depend on the rate of acceleration through this critical speed. -518V 415 and 332V I I 10 20 30 40 eccentric ity.% 50 60 Fig. 9 Variation of transient u.m.p. with eccentricity and comparison with steady-state full-load values, for 6-pole motors a Wound rotor, rotor short-circuited b Cage rotor ⢠= 518V x = 4I5V Q = 332V (i) Transient u.m.p. (ii) Steady-state full-load u.m.p., from Fig. I 1623 3.4 Effect of external rotor resistance Wound-rotor motors are normally started with external resistance in the rotor circuit, to increase the starting torque and reduce the starting current. Tests were made to find the effect of this external resistance on the magnitude of the transient u.m.p. The tests were made at each test voltage for an eccentricity of 30%, increasing the external resistance, from zero to about eight times the rotor-winding resistance. The magnitude of transient u.m.p. decreased approximately linearly as the external resistance was increased, the decrease in u.m.p. being slightly less rapid than the increase in resis- tance. For large values of external resistance, the transient u.m.p. approached the no-load value. At the rated voltage, and for the external resistance giving maximum starting torque (i.e. with the total rotor resistance equal to the rotor reactance), the transient u.m.p. was reduced to 79% of the value with a short-circuited rotor winding. The per- centage reduction in transient u.m.p., for a given external rotor resistance, increased as the voltage was reduced. Assuming a similar operation in a cage-rotor motor, it is pos- sible that the increased starting resistance of a double-cage, or deep-bar rotor, would reduce the values measured, the test motor having plain round rotor bars. 4 Comparison of u.m.p. results Table 2 shows values of no-load, full-load and transient u.m.p. for the wound-rotor and cage-rotor motors, and voltage can be found from the ratios of the per-unit values on full load to no load, these ratios being 111, 1 05 and 1 -02 at 332,415 and 518 V, respectively. Similarly, in the cage-rotor motor, u.m.p. on full load is 2 07, 1 -33 and 1 03 times the no-load values at 332, 415 and 518 V, respectively. A simple theory suggests that, owing to the greater voltage drop across the stator leakage reactance on load, the funda- mental air-gap flux will be reduced, and hence the full-load u.m.p. would be less than the no-load value. However, it is likely that the observed extra u.m.p. results from harmonic air-gap fields set up by the load currents. The large increase of u.m.p. in the cage-rotor motor, caused by load currents, can most likely be attributed to a reduction in equalising effect of the cage. This is most pronounced at reduced voltage, when the load current is greatest. The effect of the rotor slots can be seen from a comparison of values of u.m.p. for the unslotted-rotor motor and the wound-rotor motor on no load. As the flux density is main- tained constant over the stator teeth, the presence of the rotor slots reduces the u.m.p., the measured values of u.m.p. in the wound-rotor motor on no load being a mean of 90% of the values in the unslotted-rotor motor. During the transient starting conditions, the u.m.p. in the wound-rotor motor is 1 -57, 1 -35 and 1 -34 times the no-load values at 332, 415 and 518 V respectively, while the corre- sponding ratios in the cage-rotor motor are 9 0,6-7 and 4 ⢠1. Comparing per-unit values of transient u.m.p., the u.m.p. in the wound-rotor motor is 1 10, 1 01 and 1 -20 times greater than that in the cage-rotor motor. Thus, during starting, the Table 2 STEADY-STATE AND TRANSIENT U.M.P., ⬠= 1 0 % Voltage, V Measured air-gap flux density, T Measured no-load u.m.p., kN . Measured full-load u.m.p., kN . Measured transient u.m.p., kN. Rationalised no-load u.m.p., kN/T2 Rationalised full-load u.m.p., kN/T2 Rationalised transient u.m.p., kN/T2 Per-unit no-load u.m.p Per-unit full-load u.m.p Per-unit transient u.m.p Wound rotor 332 0-70 0-74 0-81 ; ( â¢15 â¢50 â¢66 >-35 )-95 â¢05 â¢49 415 0-89 0-94 0-99 1-26 119 1 -25 1 -60 0-75 0-79 101 518 103 101 102 1-33 0-94 0-95 1-25 0-59 0-60 0-79 Cage rotor 332 0-71 012 0-25 107 0-24 0-49 213 015 0-31 1 -35 415 0-87 018 0-25 119 0-24 0-32 1 58 015 0-20 100 518 I 07 0-30 0-30 119 0-26 0-26 104 016 0 1 6 0-66 Unslotted rotor 332 0-66 0-69 1-58 100 415 0-82 0 91 1 35 0-85 518 0-99 106 108 0-68 steady-state values of u.m.p. for the unslotted-rotor motor, at each test voltage for 10% eccentricity. The no-load air-gap flux density is also tabulated for each motor. it has been assumed that u.m.p. is directly proportional to the square of the air-gap flux density. This assumption has been found to be true, provided that no saturation occurs in the motor. Thus, each value of u.m.p. has been divided by the square of the no-load air-gap flux density corresponding to the particular voltage, to give a table of figures independent of the actual air-gap flux density and dependent on other variables, such as load current, saturation etc. These values have been termed rationalised values of u.m.p. The unslotted-rotor motor at 332 V under steady-state conditions is the simplest condition studied; i.e. effects of load current, saturation, rotor slotting and parallel paths in rotor and stator can be ignored. The corresponding rationalised value of u.m.p. has hence been taken as a basic per-unit value, and used to evaluate per-unit values of u.m.p. for all the test conditions. The effect of saturation on u.m.p. can be seen from the per-unit values for the unslotted-rotor motor. These three values would all be unity if no saturation occurred, but they are reduced to 0-85p.u. at 415 V and 6 68p.u. at 518 V. The effect of the cage rotor can be seen from a comparison of the per-unit no-load values for the wound-rotor and cage- rotor motors, the rotor-tooth pitch and slot openings being identical in each case. Thus, by division, the no-load u.m.p. in the cage-rotor motor is 0 1 6 , 0-20 and 0-27 times that in the wound-rotor motor at 332, 415 and 518 V, respectively. The effect of load currents in the wound-rotor motor at each 1624 cage gives virtually no equalising action, and behaves in approximately the same manner as the wound rotor. The increase of transient u.m.p. in the wound-rotor motor above no-load and full-load values contradicts the theory pre- dicting a reduction in u.m.p. due to an increased leakage- reactance voltage drop. Freise and Jordan3 derive a reduction factor from the circle diagram, which, for this machine, has the values of 0-63, 0-66, 0-70, 0 82 and 100, at standstill and speeds of 250, 500, 750 and lOOOrev/min, respectively. The transient increase of u.m.p. actually measured can, again, most likely be explained by harmonic fields. 5 5.1 Determination of u.m.p. Basic calculation Substituting values for D and / for this machine, in eqn."7 of the Appendix, gives F = 21 -75fi2e kilonewtons Hence, in the rationalised terms of Section 4, the calculated u.m.p. for 10% eccentricity is 2-175kN/T2, ignoring all effects of saturation, slotting, load currents and parallel paths. 5.2 Consideration of slotting Allowance for slotting can be made most satisfactorily by separately calculating u.m.p. over the teeth and slots, from a knowledge of the air-gap flux-density distribution. PROC. 1EE, Vol. 115, No. 11, NOVEMBER 1968 Consider a simplified version of the no-load air-gap flux density distribution of Fig. 5a, as shown in Fig. 10a. The flux densities across tooth and slot have been designated Bx in si ty fl ux d < RB1 "~lo â(10â- tooth sl ol _ position on stator surface a Fig. 10 d en si ty fl ux B2 1 ^ L-r* M tooth jfl | m ! l«n position on stator surface Simplified air-gap flux-density distribution a No load b 1Full load and RBX respectively, where R is a reduction factor, less than unity, due to the increased reluctance in the vicinity of the slot. Thus, the magnetic force newton per square metre If the stator surface were unslotted, with a constant air-gap flux density Bx, the force would be F2 = B 2/2fji0 newton per square metre Tf the same total area of air gap is considered in each case, B2(p - J) + (RBt) 2s Fl = Jp2 f If both stator and rotor surfaces are slotted, where the suffixes refer to stator and rotor, respectively. From measured air-gap flux-density distributions on no load, such as Fig. 5a, the mean ratio of flux density across the stator slots to that across the stator teeth was 0-43, for the wound-rotor, cage-rotor and unslotted-rotor motors. For simplicity, the flux density is assumed constant over the slot. As the rotor- and stator-slot openings are the same in shape and size, R2 is assumed to be equal to Rt, for the wound-rotor and cage-rotor motors, and equal to unity-for the unslotted-rotor motor. Hence, substituting in the final equation for F,, the reduction factors for u.m.p. due to slotting are 0 ⢠73 for the wound-rotor and cage-rotor motors, and 0 ⢠82 for the unslotted-rotor motor. A similar method can be used to calculate the effect of the full-load air-gap flux density distribution on u.m.p. From a simplified distribution (Fig. 106), calculation gives the ratio Full-load u.m.p. = 1 + &B2/3B2 No-load u.m.p. where AJB = increase in flux density at leading edge of tooth = decrease in flux density at trailing edge of tooth and Bx = no-load value of air-gap flux density From measurements of air-gap flux density, the ratio equalled 0-39, 0-21 arid 0 1 4 at 332, 415 and 518V, respec- tively. Substituting these values gives increases of u.m.p., due to load currents of 5 ⢠1 %, 1 ⢠5 % and 0 ⢠7 %, respectively. These values are less than the corresponding measured increases of 11%, 5 % and 2 %. However, as none of the increases is large, the errors involved are not too important. 5.3 Consideration of saturation Previous authors have allowed for saturation in two different ways. Freise and Jordan3 calculate an effective eccentricity directly from the magnetisation curve of the motor, taking the ratio of the effective eccentricity to the geometrical eccentricity as the ratio of the air-g'ap m.m.f. to PROC. IEE, Vol. 115, No. 11, NOVEMBER 1968 the total m.m.f. Similarly, Rosenberg4 and Schuisky5 derive saturation factors which are proportional to the slope of the motor magnetisation curve at the operating point. These methods all suffer by considering the overall saturation in the machine, whereas this investigation has shown that relative saturation in the large and small air gaps is very important. Saturation of the iron section of an iron-air magnetic circuit can be simply represented by assuming an increase in the length of the air section and an infinite permeability of iron. If the rotor of an induction motor is eccentrically placed in the stator bore, the air-gap flux-density distribution around the circumference of the rotor will be nonuniform. Thus, the degree of saturation, and the resulting air-gap correction, will vary around the air gap. A greater percentage increase occurs in the smaller air gap than in the larger air gap, and conse- quently the effective eccentricity is less than the geometrical eccentricity. In his thesis, Stork6 separately calculates the increase in air-gap length due 'to saturation of the core and of the teeth. Core saturation is equivalent to an effective air-gap length, given by the equation Sec o V i i ^ ji/T liAfâ Mc where ^ = geometrical air-gap length, SAf = total m.m.f. Mc = core m.m.f. The effect of tooth saturation is considered to be a wave variation of air-gap length, with a pole pitch of half the fundamental pole pitch. The mean effective air-gap length is given by the equation get ~gT,M-Mt where M, = tooth m.m.f. These two equations can be combined to calculate the effective air-gap length, due to both tooth and core saturation. For the core, A Mr gec~ g = &gc = g and for the teeth, get - S = = g SAf-Mc M, SAf- M, The total effective air gap can be found from the expression ge = g + A^c + bgl If Mc and Mt are assumed small, so that their product can be neglected with respect to the square of SM, we have ge = S M where Ma = air-gap m.m.f. Thus the effective length of air gap at any point around the rotor can be calculated if the local magnetisation curve is known. The air-gap flux densities in the wound-rotor motor for the three test values of line current can be found from Fig. 6 for a range of air-gap lengths from 0-45 to 1 10mm. Thus, for any air-gap length, a magnetisation curve can be constructed through three points, and the origin from which the effective air-gap length can be found as described above. Hence, for a given geometrical eccentricity, the effective maximum and minimum air-gap lengths and the effective eccentricity can be calculated. Using this method, the mean ratios of effective eccentricity to geometrical eccentricity were found to be 0-87, 0-71 and 0-59, at 332, 415 and 518 V, respectively, for the wound-rotor motor. Similarly, for the unslotted-rotor motor, the ratios were 0-98, 0-92 and 0-65, respectively. 5.4 Comparison of measured and calculated u.m.p. Combining the slotting and saturation factors, as previously derived, with the basic calculated u.m.p. gives rationalised values of no-load u.m.p. in the wound-rotor 1625 motor as 1 -38,113 and 0 ⢠94kN/T2, compared with measured values of 1-50, 1 1 9 and 0-94kN/T2 at 332, 415 and 518 V, respectively. These measured values are greater than those 2 0 1 8 1-6 1-4 10 0-8 a 0 6 0 4 0-2 31 kN ot518V 100 200 300 400 line voltage, V 500 600 Fig. 11 Comparison of measured values of u.m.p. with values calculated allowing for slotting and saturation, for 6-pole wound-rotor motor e =⢠10%, no load (i) Basic calculated u.m.p. (ii) U.M.P. calculated with allowance for slotting (Mi) U.M.P. calculated with allowance for slotting and saturation (iv) Measured u.m.p. ( i ) 2 0 1-8 1-6 1-4 1-2 1 0 0-8 0-6 0-4 0-2 100 200 300 400 500 600 line voltage,V 1626 calculated by a maximum of 8% at 332 V, and an average of 4 % over the range of test voltages. The quality of agreement between measured and calculated values of u.m.p. can best be seen from Figs. 11 and 12. These show the basic calculated values of u.m.p. along with values calculated with allowance for slotting and saturation, and the measured values for the wound-rotor and unslotted-rotor machines. The agreement between the final calculated u.m.p. and the measured u.m.p. can be seen to be very satisfactory, especially in relation to the differences of up to 2-3 : 1 between basic calculated values and measurements. 6 Conclusions None of the existing published theories correctly predicts u.m.p. for the 6-pole induction motor. The defects mainly result from inadequate treatment of saturation and slotting, and neglect of both equalising currents in the parallel winding paths of the cage rotor and harmonics in the air-gap field. The present work gives an adequate allowance for saturation and slotting. A simple, yet complete, theory is described and applied to the 6-pole wound-rotor motor on no load, predicting u.m.p. to within less than 10% of the measured values over a range of supply voltages, and up to 50% eccentricity. The theory agrees with measurements for the 6-pole machine, but it is likely to hold for all multipole machines with wound-rotors. Modifications will have to be made to allow similar calculations for a 2-pole machine. The calculation of full-load u.m.p. in the wound-rotor motor has not been attempted. Although simple considera- tions and experiment disagree in the direction of change of u.m.p. owing to load currents, the difference in u.m.p. between the no-load and full-load conditions is small. Calcula- tions for the no-load case should thus suffice, also, for full-load in a wound-rotor motor. The transient u.m.p. in the wound-rotor motor was on average 33 % greater than the full-load value. However, this increase during starting could be almost eliminated by means of external resistance in the rotor circuit. Assuming this method of starting the wound-rotor motor, u.m.p. would thus vary by only approximately 10% over the whole speed range. A complete theory for cage-rotor motors has not yet been developed, but data are now available on which such a theory can be based. The cage-rotor motor differs from a wound- rotor motor in allowing equalising currents in the multiple parallel paths of the cage winding. These currents have greatest effect under no-load conditions. In the tests, u.m.p. was reduced to 20% of that in the wound-rotor motor, at rated voltage. The differences diminish at full load, and almost disappear during starting. Thus, the shaft of a cage-rotor motor must be sufficiently stiff to withstand the starting u.m.p., but the bearing size could possibly be reduced to match the lower running value of u.m.p. U.M.P., during starting, has a considerable alternating component at rotor-slot frequency, which will set up resonant vibrations at a speed given by the natural resonant frequency divided by the number of rotor slots. It is possible that noise and vibration will arise from this cause in any commerical machine in its normal operating speed range. Apart from more detailed theoretical analyses, particularly of the cage-rotor motor, further experimental work will involve a study of the effect of parallel connections in the stator winding. A detailed investigation will then be made of the magnitude of u.m.p. in a 2-pole motor. All the theories suggest that such a motor is a special case, with u.m.p. less than that in a multipole machine. Fig.12 Comparison of measured values of u.m.p. with values calculated allowing for slotting and saturation for 6-pole unslotted-rotor motor e = 10%, no load (i) Basic calculated u.m.p. (ii) U.M.P. calculated with allowance for slotting (iii) U.M.P. calculated with allowance for slotting and saturation (iv) Measured u.m.p. PROC. IEE, Vol. 115, No. II, NOVEMBER 1968 7 Acknowledgments The author is grateful for numerous discussions with his colleagues- and to the Director, Electrical Research Association, for permission to publish this paper. References 1 BRADFORD, M. : 'Unbalanced magnetic pull in a 6-pole, 10 kW induction motor with a series-connected stator winding', ERA Report 5216, 1968 2 VON KAEHNE, P.: 'Unbalanced magnetic pull in rotating electric machines: Survey of published work', ERA Report Z/T142, 1963 pp. 23-24 3 FREISE, w., and JORDAN, H.: 'Einseitige magnetische Zugkrafte in Drehstrommaschinen', Elektrotech. Z., 1962, [A], 83, (9), pp. 299-303 4 ROSENBERG, E.: 'Magnetic pull in electric machines', Trans. Amer. Inst. Elect. Engrs., 1918, Pt. II, 37, pp. 1425-1469 5 SCHUISKY, w.: 'Berechnung electrischer Maschinen' (Springer Verlag, Vienna, 1960), p. 387 6 STORK, P.: 'Beitrag zur Theorie der elektromagnetischen Korper- schallerzeugung bei Drehstrom-Kafig-Laufermotoren mittlerer Leistung', Thesis, Hanover University, 1960, pp. 35-38 9 Appendix The magnetic attractive force between two plane parallel magnetic surfaces is given by a = -â (1 ) newton per square metre . . (1) where b = flux density existing between the magnetic sur- faces, T /x0 = permeability of free space or magnetic space constant = 4TT X 10~7 /x = permeability of magnetic material Fig. 13 Rotor surface For iron working at moderate levels of flux density, /x is large and, hence, to a good approximation: a = b2 newton per square metre (2) A cylindrical rotor surface is shown in Fig. 13 with a small element subtending an angle A«/» at the centre. Radial force acting on element = okA where A/4 = surface area of element = Resolving this force along the xaxis gives Fx â Ira cos Therefore, the resultant force acting on the rotor in the x direction is ⢠= â r (3) where b^ = local flux density at surface of element If the rotor is now considered to be positioned eccentrically in a stator bore, g* = gn0 - e cos ifj) (4) where g^, = air-gap length at angle )"' ~ bn(l + e cos I/J) . . . . for small eccentricities, where bn = flux density in normal air gap Therefore, Ir C2n F=â-\ b2n(\ + e cos if/) 2 cos ⢠2/^0 Jo (5) Jo irDlbh â newtons (6) where D â diameter of rotor = 2r Considering a flux wave varying sinusoidally in amplitude with time, bn can be replaced by the effective flux density, equal to { where Bp = peak air-gap flux density Thus â TTDIB2⬠F= â-â£- newtons (7) PROC. 1EE, Vol. 115, No. 11, NOVEMBER 1968 1627
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